Blend ginger, turmeric, honey, lime juice, black peppercorns, and 3 cups water in a blender until smooth. Strain through a fine-mesh sieve into a large bowl. Stir in chia seeds and let sit until seeds begin to swell, 8–10 minutes.
Divide tonic among ice-filled glasses. Top off with club soda and garnish with thyme sprigs, if desired.
Tonic can be made 1 day ahead. Cover and chill. Stir vigorously before topping off with club soda.